 Recurrent disk occupation makes an entire conformal clock infinite

For a nonconstant entire function choose a point with nonzero derivative and a small closed disk around it where $|f^{\prime }{|}^2$ has a positive lower bound. recurrence of planar Brownian motion gives infinitely many successive crossings from a concentric inner circle to the disk boundary. Their durations are positive iid variables by strong Markov and rotational invariance of Brownian motion. The strong law applied to their bounded positive truncations makes their sum infinite. Hence the conformal clock is infinite and the analytic Brownian image, under time change of a continuous process, has full infinite Brownian lifetime. Centering the disk at a noncritical point is essential.